Quantization Of Chern-Simons Theory Research Articles

The genus one complex quantum Chern–Simons representation of the mapping class group

In this paper, we compute explicitly, following Witten’s prescription, the quantum representation of the mapping class group in genus one for complex quantum Chern–Simons theory associated to any simple and simply connected complex gauge group [Formula: see text]. We use a generalization of the Weil-Gel’fand–Zak transform to exhibit an explicit expression for the representation.

Genus-one complex quantum Chern–Simons theory

We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K\{a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K\{a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.

Black Hole Entropy from Indistinguishable Quantum Geometric Excitations

In loop quantum gravity the quantum geometry of a black hole horizon consists of discrete nonperturbative quantum geometric excitations (or punctures) labeled by spins, which are responsible for the quantum area of the horizon. If these punctures are compared to a gas of particles, then the spins associated with the punctures can be viewed as single puncture area levels analogous to single particle energy levels. Consequently, if we assume these punctures to be indistinguishable, the microstate count for the horizon resembles that of Bose-Einstein counting formula for gas of particles. For the Bekenstein-Hawking area law to follow from the entropy calculation in the large area limit, the Barbero-Immirzi parameter (γ) approximately takes a constant value. As a by-product, we are able to speculate the state counting formula for the SU(2) quantum Chern-Simons theory coupled to indistinguishable sources in the weak coupling limit.

Framing anomaly in the effective theory of the fractional quantum Hall effect.

We consider the geometric part of the effective action for the fractional quantum Hall effect (FQHE). It is shown that accounting for the framing anomaly of the quantum Chern-Simons theory is essential to obtain the correct gravitational linear response functions. In the lowest order in gradients, the linear response generating functional includes Chern-Simons, Wen-Zee, and gravitational Chern-Simons terms. The latter term has a contribution from the framing anomaly which fixes the value of thermal Hall conductivity and contributes to the Hall viscosity of the FQH states on a sphere. We also discuss the effects of the framing anomaly on linear responses for non-Abelian FQH states.

Torus Knots in Lens Spaces and Topological Strings

We study the invariant of knots in lens spaces defined from quantum Chern–Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the paper, we propose a B-model topological string theory description of torus knots in L(2,1).

Quantum geometry of isolated horizons and black hole entropy

Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum Chern-Simons theory of a U(1) connection on a punctured 2-sphere, the horizon. Subtle mathematical features of the quantum Chern-Simons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (non-gravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all non-rotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper self-contained by including short reviews of the background material.

On Quantum Statistical Mechanics of a Schwarzschild Black Hole

Quantum theory of geometry, developed recently in the framework of non-perturbative quantum gravity, is used in an attempt to explain thermodynamics of Schwarzschild black holes on the basis of a microscopical (quantum) description of the system. We work with the formulation of thermodynamics in which the black hole is enclosed by a spherical surface B and a macroscopic state of the system is specified by two parameters: the area of the boundary surface and a quasilocal energy contained within. To derive thermodynamical properties of the system from its microscopics we use the standard statistical mechanical method of Gibbs. Under a certain number of assumptions on the quantum behavior of the system, we find that its microscopic (quantum) states are described by states of quantum Chern-Simons theory defined by sets of points on B labelled with spins. The level of the Chern-Simons theory turns out to be proportional to the horizon area of black hole measured in Planck units. The statistical mechanical analysis turns out to be especially simple in the case when the entire interior of B is occupied by a black hole. We find in this case that the entropy contained within B, that is, the black hole entropy, is proportional to the horizon surface area.

CHERN–SIMONS APPROACH TO THREE-MANIFOLD INVARIANTS

A new, formal, noncombinatorial approach to invariants of three-dimensional manifolds of Reshetikhin, Turaev and Witten in the framework of nonperturbative topological quantum Chern–Simons theory, corresponding to an arbitrary compact simple Lie group, is presented. A direct implementation of surgery instructions in the context of quantum field theory is proposed. An explicit form of the specialization of the invariant to the group SU(2) is shown.

THE QUANTUM GROUP REALIZATIONS OF 3-D CHERN–SIMONS THEORIES

The algebra of the integrated connections and of their traces is considered in the one-genus sector of classical and quantum Chern–Simons theory. In the classical case this algebra is braid-like and although the corresponding Jacobi identities are satisfied, the associated r-matrix does not satisfy the classical Yang–Baxter equations. However, it turns out this algebra originates a "quantum" algebra SU (2)q given by its trace algebra. Canonical quantization of the above algebra is performed and a one-parameter expression for the operator ordering is considered. The same quantum algebra with a modified deformation parameter, nontrivially depending on ħ, is obtained.

Chern-Simons theory as the large-mass limit of topologically massive Yang-Mills theory

We study quantum Chern-Simons theory as the large-mass limit of the limit D→3 of dimensionally regularized topologically massive Yang-Mills theory. This approach can also be interpreted as a BRS-invariant hybrid regularization of Chern-Simons theory, consisting of a higher-covariant derivative Yang-Mills term plus dimensional regularization. Working in the Landau gauge, we compute radiative corrections up to second order in perturbation theory and show that there is no two-loop correction to the one-loop shift k→ k+ c v, k being the bare Chern-Simons parameter. In passing, we also prove by explicit computation that topologically massive Yang-Mills theory is UV finite.

Universality in quantum Hall systems

It is shown that the theory of the quantum Hall effect is closely related to Chern-Simons gauge theory and to rational conformal field theory. In particular, the equations of classical electromagnetism in quantum Hall systems are derived from a pure Chern-Simons action. This permits us to argue that the scaling limit of a quantum Hall system is described by quantum Chern-Simons theory, thereby explaining the discreteness of the set of values of the Hall conductivity. Using the equivalence of Chern-Simons theory to chiral current algebra, a general derivation of the existence of chiral currents circulating near the boundary edges of a quantum Hall system is found. It is indicated how to derive the effective Chern-Simons action of a quantum Hall system from the underlying quantum mechanics. Applications of our results to very weakly doped “high-temperature superconductors” are described.

Chern-Simons and conformal field theories

The quantization of Chern-Simons theory within the context of the operator formalism is discussed. Fusion rules and modular transformations of characters of rational conformal field theories are obtained from the properties of the resulting Hilbert space.